Casimir-Foucault interaction: Free energy and entropy at low temperature

It was recently found that thermodynamic anomalies which arise in the Casimir effect between metals described by the Drude model can be attributed to the interaction of fluctuating Foucault (or eddy) currents [F. Intravaia and C. Henkel, Phys. Rev. Lett. 103, 130405 (2009).] We focus on the transverse electric (TE) polarization, where the anomalies occur, and show explicitly that the two leading terms of the low-temperature correction to the Casimir free energy of interaction between two plates are identical to those pertaining to the Foucault current interaction alone, up to a correction which is very small for good metals. Moreover, a mode density along real frequencies is introduced, showing that the TE contribution to the Casimir free energy, as given by the Lifshitz theory, separates in a natural manner into contributions from eddy currents and propagating cavity modes, respectively. The latter have long been known to be of little importance to the low-temperature Casimir anomalies. This convincingly demonstrates that eddy current modes are responsible for the large temperature correction to the Casimir effect between Drude metals, predicted by the Lifshitz theory, but not observed in experiments.